Question: The sum of two numbers is $83$, and their difference is $27$. What are the two numbers?
Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 83}$ ${x-y = 27}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 110 $ $ x = \dfrac{110}{2} $ ${x = 55}$ Now that you know ${x = 55}$ , plug it back into $ {x+y = 83}$ to find $y$ ${(55)}{ + y = 83}$ ${y = 28}$ You can also plug ${x = 55}$ into $ {x-y = 27}$ and get the same answer for $y$ ${(55)}{ - y = 27}$ ${y = 28}$ Therefore, the larger number is $55$, and the smaller number is $28$.